The equation of a tangent to the parabola x^2=8y, which makes an angle with the positive d…

Question

The equation of a tangent to the parabola $x^2=8y$, which makes an angle $	heta$ with the positive direction of the $x$-axis, is

Accepted Answer

Parametrize $x=4t,y=2t^2$; slope $\dfrac{dy}{dx}=\dfrac x4=t$. Tangent: $y=tx-2t^2$; setting $t=\tan\theta$:
$$y=x\tan\theta-2\tan^2\theta\ \Longrightarrow\ x=\frac{y+2\tan^2\theta}{\tan\theta}=y\cot\theta+2\tan\theta.$$

Solutions · 1